A satellite orbiting Earth at an orbital radius r has a velocity v. What would the velocity be if the satellite is moved to an orbital radius of 6r?(1 point)

Question

Responses

16√v
the fraction with numerator 1 and denominator square root of 6 v

16v
1 sixth v

6–√v
square root of 6 v

6v

Answer 1

To determine the new velocity of a satellite when it is moved to an orbital radius of \(6r\), we can use the formula for the orbital velocity:

\[ v = \sqrt{\frac{GM}{r}} \]

where:

• \(G\) is the gravitational constant,
• \(M\) is the mass of the Earth,
• \(r\) is the orbital radius.

If the original orbital radius is \(r\) and the original velocity is \(v\), we can express \(v\) as:

\[ v = \sqrt{\frac{GM}{r}} \]

For the new orbital radius of \(6r\), the new velocity \(v'\) can be expressed as:

\[ v' = \sqrt{\frac{GM}{6r}} \]

We can relate \(v'\) to \(v\):

\[ v' = \sqrt{\frac{GM}{6r}} = \sqrt{\frac{1}{6}} \cdot \sqrt{\frac{GM}{r}} = \sqrt{\frac{1}{6}} v \]

Thus, we can write:

\[ v' = \frac{v}{\sqrt{6}} \]

Among the provided options, this corresponds to:

the fraction with numerator 1 and denominator square root of 6 \(v\).